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REDUCING THE NUMBER OF FIXED POINTS OF SOME HOMEOMORPHISMS ON NONPRIME 3-MANIFOLDS XUEZHI ZHAO Received 5 September 2004; Revised 15 March 2005; Accepted 21 July 2005 We will consider the number of fixed points of homeomorphisms composed of finitely many slide homeomorphisms on closed oriented nonprime 3-manifolds. By isotoping such homeomorphisms, we try to reduce their fixed point numbers. The numbers ob- tained are determined by the intersection information of sliding spheres and sliding paths of the slide homeomorphisms involved. Copyright © 2006 Xuezhi Zhao. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Nielsen fixed point theory (see [1, 4]) deals with the estimation of the number of fixed points of maps in the homotopy class of any given map f : X → X. The Nielsen number N( f ) provides a lower bound. A classical result in Nielsen fixed point theor y is: any map f : X → X is homotopic to a map with exactly N( f ) fixed points if the compact polyhe- dron X has no local cut point and is not a 2-manifold. This includes all smooth manifolds with dimension g reater than 2. It is also an interesting question whether the Nielsen number can be realized as the number of fixed points of a homeomorphism in the isotopy class of a given homeomor- phism. In fact, it is just what J. Nielsen expected when he introduced the invariant N( f ). Assume that X is a closed manifold. The answer to this question is obviously positive for the unique closed 1-manifold. A positive answer was given by Jiang and Guo [5]for 2-manifolds, and was given by Kelly [7] for manifolds of dimension at least 5. In [6], Jiang, Wang and Wu proved that for any closed oriented 3-manifold X which is either Haken or geometric, any orientation-preserving homeomorphism f : X → X is isotopic to a homeomorphism with N( f ) fixed points ([6, Theorem 9.1]). If Thurston’s geometric conjecture is true, all nonprime 3-manifolds are of this type. In this paper, we will consider a certain class of homeomorphisms of closed, oriented 3-manifolds that have a connected sum decomposition into prime factors, namely irre- ducible manifolds and copies of S 2 × S 1 , and at least two factors (nonprime manifolds). Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 25897, Pages 1–19 DOI 10.1155/FPTA/2006/25897 2 Fixed points of slide homeomorphisms It is known from work of Kneser and Milnor that in the oriented setting, the pr ime and irreducible factors of the decomposition are unique. We examine homeomorphisms that can be expressed as the composition of finitely many slide homeomorphisms. A so-called slide homeomorphism is the identity away from a certain stratified open neighborhood, the sliding set, of a torus, and is defined by a family of rotation-like transformations on this set. According to McCullough’s result (see [8]), an arbitrary homeomorphism of a re- ducible 3-manifold can be expressed as the composition of homeomorphisms that comes in four types, one of which is that of slide homeomorphisms. In [9], the author considered the Nielsen numbers and fixed points of homeomor- phisms which are compositions of m slide homeomorphisms on nonprime 3-manifolds. The fixed point index of the complement of the union of the sliding sets was proved to be zero. When m = 2, we found presentations that are in some sense “standard,” for which the fixed point numbers, the fixed point class coordinates and the fixed point indices for all fixed points can be determined. Thus, we were able to give some estimating b ounds on the Nielsen numbers of such kinds of homeomorphisms. The present paper is a continu- ation of [9]. We will generalize the results for m = 2 there to the case where m can be an arbitr ary positive integer. We will focus on a geometrical method to reduce the number of fixed points in any given isotopy class of such a homeomorphism. The lower bound prop- erty of Nielsen number implies that our number of fixed points yields an upper bound for Nielsen number. The remaining sections are organized as follows. In Section 2, we will fix notation which will be used throughout this paper, and recall the definition of slide homeomor- phism. In Section 3, we will show (Lemma 3.4) that away from the sliding set, f can be isotoped to a fixed point free homeomorphism by an arbitrary small isotopy. Although each component of this set has zero fixed point index ([9, Theorem 3.2]), the result here is not very obvious because we are considering fixed points up to isotopy rather than homotopy. In Section 4, fixed points over the sliding set are considered. It is ar- gued that f is isotopic to a homeomorphism with finitely many fixed points, and that the size of this fixed point set is expressible in terms self-intersection data for the slid- ing set (Proposition 4.6). Reducing the number of fixed points for homeomorphisms in the isotopy class of f then involves controlling in some sense the number of self inter- sections; our main result (Theorem 4.11) gives a lower bound for this number. Finally, a short Section 5 shows that in some cases, one may simplify and “optimize” the sliding set so that the bound in Section 4 can be further lowered, that is, the number of fixed points can be further reduced. 2. Conventions and notations In this section, we will make necessar y conventions in notation, which will be used in later sections. (1) The underlying manifold M. In this paper, the manifold M is assumed to b e a closed oriented 3-manifold, which is nonprime. It is known that M can be written as a connected sum of finitely many prime 3-manifolds, that is, M = M 1 #M 2 #···#M n  #···#M n  +n  ,in which M i is irreducible for 1 ≤ i ≤ n  and M i = S 2 × S 1 for n  +1≤ i ≤ n  + n  . The non- prime property implies that n  + n  > 1. Xuezhi Zhao 3 Take a 3-s phe re and re move n  +2n  open discs to obtain a punctured 3-cell W with n  +2n  boundary components. We then have that M = W ∪ (∪ n  +n  i=1 M  i ), where M  i = M i − Int(D i )for1≤ i ≤ n  and M  i = S 2 × I for n  +1≤ i ≤ n  + n  (see [8]). Each M  i admits the orientation coincident with that of M, and each ∂M  i inherits the orientation of M  i . (2) Slide homeomorphisms.LetS be an oriented essential 2-sphere in M, which is orientation-preservingly isotopic to a boundary component of ∂M  j .Letα : I → M be a path without self intersection in M such that α ∩ M  j = α ∩ S ={α(0),α(1)}.Taketwo regular neighborhoods N  and N  (N  ⊂ Int(N  )) of α ∪ S in M.ThenInt(N  − N  )has two components which are homeomorphic to S 2 × (0,1) and T 2 × (0,1) respectively. We write the latter as T(S,α). Pick a coordinate function c : T(S,α) → T 2 × (0,1), where the points in T 2 × (0,1) are labeled by (θ,ϕ,t), such that the θ-line, c −1 (θ,∗,∗), is parallel to the oriented path α and the t-line c −1 (∗,∗,t) m oves radially away from the path α when the value of t is increased. A slide homeomorphism s : M → M determined by α and S is defined by s(x) = ⎧ ⎨ ⎩ c −1 (θ +2πt,ϕ,t)ifx = c −1 (θ,ϕ,t) ∈ T(S,α), x otherwise, (2.1) denoted by s(S,α). The sets T(S,α), S and α are said to be respectively the sliding set, sliding sphere and sliding path of s(S,α). (3) Orientations and isotopies. Since all manifolds under consideration are oriented, including sliding spheres and sliding paths, isotopies here are considered to be ambient and orientation-preserving. For example, if M = M 1 #M 2 isaconnectedsumoftwoprime manifolds, ∂M  1 and ∂M  2 are not regarded as isotopic. (4) Fundamental groups and path classes. Consider the construction M =W∪(∪ n  +n  i=1 M  i ) of M.Wechooseapointx 0 in W as its base point. To any path γ with ending points in W there corresponds uniquely an element γ ∗ γγ −1 ∗∗  in π 1 (M,x 0 ), where γ ∗ and γ ∗∗ are path from x 0 to γ(0) and γ(1) in W respectively. By abuse of notation, we write it simply as γ.Choosex j ∈ ∂M  j as base point of M  j for j = 1,2, ,n  + n  .Thus,eachπ 1 (M  j ,x j )is embedded into π 1 (M,x 0 ) in a natural way as above, and hence π 1 (M,x 0 )isthefreeprod- uct of π 1 (M  j ,x j ), j = 1,2, ,n  + n  . We write simply as π 1 (M,x 0 ) = π 1 (M  1 ) ∗ π 1 (M  2 ) ∗ ···∗ π 1 (M  n  +n  ), which is also equal to π 1 (M 1 ) ∗ π 1 (M 2 ) ∗···∗π 1 (M n  +n  ). (5) The homeomorphism f .Fromnowon, f is assumed to be a homeomorphism com- posed of finitely many slide homeomorphisms, that is, f = s(S m ,α m ) ◦ s(S m−1 ,α m−1 ) ◦ ···◦ s(S 1 ,α 1 ). The union ∪ m j =1 T(S j ,α j ) of all sliding sets is said to the sliding set of f .For a simplification in notation, we write s m  ···m  for the composition s(S m  ,α m  ) ◦ s(S m  −1 , α m  −1 ) ◦···◦s(S m  ,α m  )foranym  and m  (1 ≤ m  <m  ≤ m). In particular, s(S j ,α j )is simply written as s j . (6) General position. Consider the slide homeomorphisms whose composition is f .We can ensure the sliding paths α 1 ,α 2 , ,α m , a nd sliding spheres S 1 ,S 2 , ,S m areingeneral position relative to the set ∪ m j =1 {α j (0),α j (1)}. Thus, these sliding paths have no intersec- tion, and α i intersects with S j transversally for i = j. Since each sliding sphere is isotopic to a component of −∂W, we can arrange these sliding spheres to be disjoint. In this sit- uation, if each sliding set T(S j ,α j ) is in a small neig hborhood of α j ∪ S j ,thenumber 4 Fixed points of slide homeomorphisms α i q (i,j;1) q (i,j;2) S j ······ Figure 2.1 of components of intersection of two sliding sets T(S j  ,α j  )andT(S j  ,α j  ) is equal to the number of points in (α j  ∪ S j  ) ∩ (α j  ∪ S j  )forall j  and j  with j  = j  . In this situation, we say that the sliding set ∪ m j =1 T(S j ,α j )of f is in general position. (7) Components B (∗,∗;∗) of the intersection of sliding sets. If the sliding set ∪ m j =1 T(S j ,α j ) is in general position, the points in α i ∩ S j (i = j) are denoted by q (i, j;1) ,q (i, j;2) , , q (i, j;|α i ∩S j |) (see Figure 2.1), where the last subscript indicates the order in α i ∩ S j along the direction of α i , that is, α −1 i (q (i, j;k  ) ) <α −1 i (q (i, j;k  ) )inI = [0,1]ifandonlyifk  < k  . The corresponding components of T(S i ,α i ) ∩ T(S j ,α j ) nearby are written as B (i, j;1) , B (i, j;2) , ,B (i, j;|α i ∩S j |) .Obviously,wehave Proposition 2.1. If the sliding set ∪ m j =1 T(S j ,α j ) is in general position, then each B (∗,∗;∗) is homeomorphic to a solid torus, and T(S i ,α i ) ∩ T(S j ,α j ) = ( |α i ∩S j | k=1 B (i, j;k) )  ( |α j ∩S i | l=1 B (j,i;l) ) for any i and j,wherei, j = 1,2, ,m with i = j. 3. Removing fixed points on the complement of sliding set Consider our homeomorphism f . Since the fixed point set of each slide homeomorphsim s i is just M − T(S i ,α i ), the points in the complement M −∪ m j =1 T(S j ,α j ) of the sliding set of f are totally contained in the fixed p oint set of f .In[9], we proved that this isolated fixed point set has zero fixed point index. In this section, we will show that this fixed point set can be removed by arbitrary small isotopy. The following definition is originally from [2]. Definit ion 3.1. Let Γ : N → TN be a vector field on a compact smooth n-manifold N. The manifold N is said to be a manifold with corners for the vector field Γ if Γ has no singular points on ∂N and if ∂N can be considered a union of (n − 1)-manifolds (with boundaries) ∂ o N, ∂ + N and ∂ − N with ∂ + N ∩ ∂ − N =∅such that Γ(x)istangentto∂ o N for x ∈ ∂ o N, points inward to N for x ∈ ∂ − N and points outward from N for x ∈ ∂ + N. Clearly, for an n-manifold N with corners, both of ∂ + N ∩ ∂ o N and ∂ − N ∩ ∂ o N are (n − 2)-dimensional closed manifolds. A simple example is the following. Xuezhi Zhao 5 Example 3.2. A constant vector field on R 2 is given by Γ(x, y) = (1, 0). Then the subset N = [0,1] × [0,1] is a manifold with corners for such a vector field Γ,with∂ o N = [0,1] × { 0,1}, ∂ − N ={0}×[0,1] and ∂ + N ={1}×[0,1]. The next lemma is a kind of generalization of the Poincar ´ e-Hopf vector field index theorem. There are some similar statements in dynamical system theory, see for example [3, Lemma A.1.3]. Lemma 3.3. Let N be a 3-manifold with corners for a vector field Γ.Iftheboundary∂N is a disjoint union of m-copies of a sphere such that ∂ o N is a disjoint union of m-copies of an annulus, and either ∂ + N or ∂ − N is a disjoint union of m-copies of a disc, then we can change Γ relative to a neighborhood of ∂N in N into a nonsingular vector field Γ  . Proof. Through a coordinate function, each component of ∂N canberegardedasoneof the following: C k =  (x, y,z):|x|≤4, (y − 8k) 2 + z 2 = 4orx =±4, (y − 8k) 2 + z 2 ≤ 4  , (3.1) where k = 1,2, ,m.Since∂ + N ∩ ∂ − N =∅, we may assume that ∂ o N =∪ m k =1  (x, y,z):|x|≤4, (y − 8k) 2 + z 2 = 4  , ∂ − N =∪ m k =1  (x, y,z):x = 4, (y − 8k) 2 + z 2 ≤ 4  , ∂ + N =∪ m k =1  (x, y,z):x =−4, (y − 8k) 2 + z 2 ≤ 4  . (3.2) Regard a neighborhood of ∂N as a subset outside of the cylinders: D k =  (x, y,z):|x|≤4, (y − 8k) 2 + z 2 ≤ 4  , k = 1,2, ,m. (3.3) Since N is a manifold with corners for the vector field Γ, Γ points inward for the cylinders (outward for N)at∂ + N and points outward for the cylinders (inward for N)at∂ − N.We have that Γ(p) ∈{(x, y,z):x>0} for p ∈ ∂ + N ∪ ∂ − N. It is not difficult to prove that the restriction of Γ at each component of ∂N, as a map from a sphere to R 3 −{0},haszero degree. Hence, we can extend Γ to the union ∪ m k =1 D k such that there is no singular point on ∪ m k =1 D k . Since N ∪ (∪ m k =1 D k ) is a closed 3-manifold, its Euler characteristic number is zero. Using standard methods in differential topology, we can deform Γ into a nonsingular Γ  relative to a neighborhood of ∪ m k =1 D k .ThenΓ  | N is our desired vector field.  Using this lemma, we can prove the following lemma. 6 Fixed points of slide homeomorphisms Lemma 3.4. Assume that the sliding set of f is in general position. Given any positive number ε, there is an isotopy F : M × I → M from f to f  satisfying: (i) d(F(x,t), f (x)) <εfor any x ∈ M and any t ∈ I, (ii) the support set {x ∈ M : F(x,t) = f (x) for some t ∈ I} of F is contained in the ε- neighborhood N ε (M −∪ m j =1 T(S j ,α j )) of the complement of the sliding set ∪ m j =1 T(S j ,α j ) in M, (iii) Fix( f  ) = Fix( f )− (M −∪ m j =1 T(S j ,α j )). Proof. Clearly, we can regard a neighborhood N(∂W)of∂W in M asasubsetofR 3 so that ∂W =−∪ n  +2n  j=1 C j ,where C j =  (x, y,z) ∈ R 3 :(x, y,z):|x|≤4, (y − 8j) 2 + z 2 = 4 or x =±4, (y − 8 j) 2 + z 2 ≤ 4  , (3.4) having the orientation induced from R 3 .Since∂W =−∪ n  +n  j=1 ∂M  j ,wemayarrangeso that ∂M  j = C j for 1 ≤ j ≤ n  ; ∂M  n  +j = C n  +2j−1 ∪ C n  +2j . The set W is located outside of these C j ’s with respect to the given orientation of C j ’s. Clearly, we can construct a vector field Γ 0 : M → TM on M so that Γ 0 (p) ={1,0,0} for any p in the neighborhood N(∂W)of∂W in M,where N(∂W) =∪ n  +2n  k=1  (x, y,z) ∈ R 3 :(x, y,z):|x|≤3, 1 ≤ (y − 8k) 2 + z 2 ≤ 9 or 3 ≤|x|≤5, (y − 8k) 2 + z 2 ≤ 9  . (3.5) Thus, W and all M  j ’s are manifolds with corners for Γ 0 .ApplyLemma 3.3 to W and all M  j ’s, we will get a nonsingular vector field Γ : M → TM on M so that Γ(p) ={1,0,0} for any p ∈ N(∂W). By definition of slide homeomorphism, each sliding sphere S k is isotopic to a C j in M. We then have a well-defined correspondence μ : {1,2, ,m}→{1,2, ,n  +2n  } such that S k is isotopic to C μ(k) in M for any k = 1,2, ,m. We take S k to be the sphere outside of C μ(k) by a distance of ν k (0 < ν k < 1). Moreover, we can arrange these ν 1 ,ν 2 , ,ν m to have distinct values. Each sliding path α k attaches the corresponding sliding sphere S k at “top” and “bottom” perpendicularly. More precisely, α k (u) = (ν k ,8μ(k),2 + ν k + u)andα k (1 − u) = (ν k ,8μ(k),−2 − ν k +1− u)forsmallu ∈ I.Eachpointq (j,k;∗) in α j ∩ S k lies on (x(q (j,k;∗) ),8μ(k)+2+ν k ,0), where all possible x(q (j,k;∗) ) are distinct numbers in (−1,1) (see Figure 3.1). We can make such an arrangement because any two sliding spheres and any two slid- ing paths have no intersection by the general position assumption. We then arrange the sliding set to lie in a sufficiently small neighborhood of ∪ m j =1 (α j ∪ S j ). Let ξ : M × R → M be the flow generated by Γ. We will show that ξ( f (p),t) = p for all points p in a small η-neighborhood N η of M −∪ m j =1 T(S j ,α j )inM provided t is small enough. Case 1. If p ∈ M −∪ m i =1 T(S i ,α i ), then f (p) = p.SinceΓ has no zero, we have that ξ( f (p), t) = ξ(p,t) = p when t is small enough. Xuezhi Zhao 7 z x y α k (1) S k α j  q ( j  ,k;∗) q ( j,k;∗) α j α k (0) Figure 3.1 Case 2. If p ∈∪ m i =1 T(S i ,α i ), then there is a unique smallest number j with p ∈ T(S j ,α j ). There are two subcases. Subcase 2.1. If s j (p) ∈∪ m i = j+1 T(S i ,α i ), then f (p) = s j (p). By general position, we can ar- range α j so that Γ(α j (u)) does not parallel to the tangent vector of α j (u)atu for all u ∈ I. Thus, ξ( ·,t) will not push along (or opposite) to the direction that s j does. It follows that ξ( f (p),t) = p when p is closed to the boundary ∂T(S j ,α j )ofT(S j ,α j ) (see Figure 3.2). Subcase 2.2. If s j (p) ∈∪ m i = j+1 T(S i ,α i ), then there is a unique smallest number k with k>jsuch that s j (p) ∈ T(S k ,α k ). Notice that p is close to ∂(∪ m i =1 T(S i ,α i )). We have that s j (p) is also close to ∂T(S k ,α k ) because the difference between p and s j (p)issmall,so s k ◦ s j (p) will not meet any sliding set other than T(S k ,α k )andT(S j ,α j ). It follows that f (p) = s k ◦ s j (p). The component of T(S k ,α k ) ∩ T(S j ,α j )aroundp and f (p)havetwotypes:B (k, j;∗) and B (j,k;∗) . In the first type, we explain the behavior of ξ( f (p),t)intwopartsofFigure 3.3. The first two stages from p to s k ◦ s j (p) is shown on the left part. The last stage is illus- trated in the right par t, where s j (p)isbehind f (p) = s k ◦ s j (p). Let p = (x p , y p ,z p ), we have  x p , y p ,z p  s j  x p , y  p ,z  p  s k  x p , y  p ,z  p  ξ(·,t)  x  p , y  p ,z  p  . (3.6) This implies that in R 3 , p and ξ( f (p),t)willhavedifferent x-values when t is small enough. It follows that ξ( f (p),t) = p.TheproofforthetypeB (j,k;∗) is the same. DefineanisotopyF δ,η : M × I → M by F δ,η (p,t) = ⎧ ⎪ ⎨ ⎪ ⎩ ξ(p,δt)ifp ∈ M −∪ m j =1 T  S j ,α j  , ξ  p,max  η − d  p,∪ m j =1 ∂T  S j ,α j  ,0  δt  if p ∈∪ m j =1 T  S j ,α j  . (3.7) 8 Fixed points of slide homeomorphisms ps j (p) = f (p) T(S j ,α j ) ξ( f (p),t) α j ξ(·,t) Figure 3.2 T(S j ,α j ) S j p T(S k ,α k ) s k ◦s j (p) s j (p) z y α k T(S j ,α j ) p ξ( f (p),t) s k ◦s j (p) T(S k ,α k ) z x Figure 3.3 Note that the arguments for ξ still work for F δ,η ,sowecanprovethatF δ,η ( f (p), t) = p for all t ∈ I and p in the η-neighborhood N η of M −∪ m j =1 T(S j ,α j )inM.Thus,whenδ and η are small enough, F δ,η will be a desired isotopy.  Corollary 3.5. Any slide homeomorphism is isotopic to a fixed point free map. 4. Fixed points on sliding sets In this section, we try to reduce the fixed points of the homeomorphism f on its slid- ing set ∪ m j =1 T(S j ,α j ). For an arbitrary fixed point x of f on its sliding set, we exam- ine its “trace” x,s 1 (x), s 21 (x), ,s m···1 (x) under the sliding homeomorphisms composing f . Lemma 4.1 will show that the sliding sets of individual slide homeomorphism meet- ing this trace is totally determined by x itself provided that each sliding set T(S j ,α j )is small enough. Hence, a fixed point x will determine a unique sequence consisting of the Xuezhi Zhao 9 components of the intersection of sliding sets, which we call the accompanying sequence (Proposition 4.2). All the possible accompanying sequence will be given in Lemma 4.3. Next, we will isotope the g iven homeomor phism f so that different fixed points on slid- ing set of f have different accompanying sequences (Lemma 4.4). When the sliding set of f is in general position, there is a unique point (α i ∪ S i ) ∩ (α j ∪ S j ) near an arbitrary component of T(S i ,α i ) ∩ T(S j ,α j ). Thus, in some sense, reducing the number of fixed points is equivalent to reducing the number of intersection points between the sliding paths and sliding spheres. The minimal number MI( {α 1 , ,α m },{S 1 , ,S m }) of the in- tersection of sliding paths and sliding spheres gives a possible number of fixed points for homeomorphisms in the isotopy class of f (Theorem 4.11). Since the Nielsen number N( f ) is a lower bound of the number of fixed points for maps in the homotopy class of f , the minimal number MI( {α 1 , ,α m },{S 1 , ,S m }) also provides an upper bound of N( f ). Lemma 4.1. If any three of these sliding sets T(S j ,α j )’s have no common points, then to each fixed point x of f there is associated a unique sub-sequence {i 1 ,i 2 , ,i k } of {1,2, ,m} with k ≥ 2 such that s i k ◦···◦s i 2 ◦ s i 1 (x) = x ∈ T(S i 1 ,α i 1 ),andsuchthats i j−1 ◦···◦s i 2 ◦ s i 1 (x) ∈ T(S i j ,α i j ) for j = 2,3, ,k. Proof. Let x be a fixed point of f in ∪ m i =1 T(S i ,α i ). There is a unique minimal i such that x ∈ T(S i ,α i ). We write this number as i 1 .Asequence{i 1 ,i 2 , ,i k } will be defined induc- tively: i j = min  n : n>i j−1 , s i j−1 ◦···◦s i 1 (x) ∈ T  S j ,α j  . (4.1) Since x ∈ T(S i 1 ,α i 1 ), we have s i 1 (x) = x.Iftherewasnosuchanumberi 2 , s i 1 (x) ∈ T(S i ,α i ) for all i>i 1 .Thus, f (x) = s m···1 (x) = s m···i 1 (x) = s i 1 (x). This would contradict the fact that x is a fixed point of f ,sowealwayshavethatk ≥ 2. By definition of i j ,wehavethats n···1 (x) = s i 1 (x)ifi 1 ≤ n<i 2 , and that s n···1 (x) = s i 2 ◦ s i 1 (x)ifi 2 ≤ n<i 3 . Inductively, we will get that s n···1 (x) = s i p−1 ◦···◦s i 2 ◦ s i 1 (x)if i p−1 ≤ n<i p . When our induction stops at a stage i p ,wehavethats i p ···1 (x) = s i p ◦···◦s i 2 ◦ s i 1 (x) does not lie in any sliding set T(S n ,α n )withn>i p ,sos m···i p ◦ s i p−1 ◦···◦s i 1 (x) = s i p ◦ s i p−1 ◦···◦s i 1 (x). It follows that f (x) = s m···1 (x) = s i p ◦ s i p−1 ◦···◦s i 1 (x). This point is just x because x is a fixed point of f . Thus, this i p is the final number, say i k ,inour subsequence of {1,2, ,m}. Let us prove the uniqueness of such a subsequence. If there is another subsequence { j 1 , j 2 , , j l } satisfying the same conditions as {i 1 ,i 2 , ,i k }, then we wil l get that x ∈ T(S j 1 ,α j 1 ). Since any three of the sliding sets have no common points, j 1 is equal to either i 1 or i k . If the last case happens, that is, j 1 = i k , by the choice of i k ,wehavethats j (x) = x for all j>i k . Thus, there would be no j 2 . It follows that i 1 = j 1 . Assume that j p = i p for p = 1,2, ,n − 1. By the property of the subsequence {i 1 ,i 2 , , i k },wehaves i n−1 ◦ ◦ s i 1 (x) ∈T(S i n ,α i n ); by the property of the subsequence { j 1 , j 2 , , j l }, we hav e s j n−1 ◦ ◦ s j 1 (x) ∈ T(S j n ,α j n ). Our assumption implies that s i n−1 ◦ ◦ s i 1 (x)and s j n−1 ◦ ◦ s j 1 (x) are the same point. Since this point lies in the image of T(S i n−1 ,α i n−1 ) = T(S j n−1 ,α j n−1 ) under homeomorphism s i n−1 = s j n−1 .ItalsoliesinT(S i n−1 ,α i n−1 ). Since any 10 Fixed points of slide homeomorphisms three of the sliding sets have no common points, T(S i n ,α i n ), T(S j n ,α j n )andT(S i n−1 ,α i n−1 ) are at most two different sets. Because i n = i n−1 and j n = j n−1 = i n−1 , the unique possibil- ity is that j n = i n . Thus, we can prove by induction that j n = i n for n = 1,2, ,min{k,l}. It remains to show that k = l.Ifk<l, then from the property of the subsequence { j 1 , j 2 , , j l },wehavethats j k ◦···◦s j 1 (x) ∈ T(S j k+1 ,α j k+1 ).Sincewehaveprovedthat j n = i n for n = 1,2, ,k, s j k ◦···◦s j 1 (x) = s i k ◦···◦s i 1 (x) = x.Thus,x lies in T(S i 1 ,α i 1 ) ∩ T(S i k ,α i k ) ∩ T(S j k+1 ,α j k+1 ). Since i k >i 1 , j k+1 is equal to either i 1 = j 1 or i k = j k . This is a contradiction. Symmetrically, the case k>lcannot happen.  For such a fixed point x,wewriteB 1 for the component of T(S i k ,α i k ) ∩ T(S i 1 ,α i 1 )con- taining x,andwriteB j , j = 2,3, ,k, for the component of T(S i j−1 ,α i j−1 ) ∩ T(S i j ,α i j ) containing s i j−1 ◦···◦s i 2 ◦ s i 1 (x). The sequence {B 1 ,B 2 , ,B k } is said to be the accom- panying sequence of x in the components of the intersection of sliding sets. Clearly, the set {B 1 ,B 2 , ,B k } itself is just the set of all components of the intersection of sliding sets containing s n···1 (x)forsomen. In other words, we have Proposition 4.2. Let x beafixedpointof f and {i 1 ,i 2 , ,i k } be its associated sub-sequence of {1,2, ,m}.Let{B 1 ,B 2 , ,B k } be a set consisting of some components of the intersection of sliding sets such that B 1 is a component of T(S i k ,α i k ) ∩ T(S i 1 ,α i 1 ),andsuchthatB j , j = 2,3, ,k,isacomponentofT(S i j−1 ,α i j−1 ) ∩ T(S i j ,α i j ). Assume that any three of these sliding sets have no common points. Then, {B 1 ,B 2 , ,B k } is the accompanying sequence of the fixed point x of f if and only if x belongs to the following set: s i k ◦ s i k−1 ◦···◦s i 1 (B 1 ) ∩ s i k ◦ s i k−1 ◦···◦s i 2 (B 2 ) ∩···∩s i k (B k ) ∩ B 1 . (4.2) Lemma 4.3. Assume that the sliding set ∪ m j =1 T(S j ,α j ) of f is in general position. If s j (B (i, j;k) ) ∩ B (i  , j;k  ) =∅unless i = i  and k = k  , the n the accompanying sequence of each fixed point of f in sliding sets has either one of the following forms:  B (i k ,i 1 ;∗) ,B (i 1 ,i 2 ;∗) ,B (i 2 ,i 3 ;∗) , ,B (i k−1 ,i k ;∗)  ,  B (i 1 ,i k ;∗) ,B (i 2 ,i 1 ;∗) ,B (i 3 ,i 2 ;∗) , ,B (i k ,i k−1 ;∗)  , (4.3) where 1 ≤ i 1 <i 2 < ···<i k ≤ m (see Figure 4.1). Proof. Let x be a fixed point of f in the sliding set with accompanying sequence {B 1 , B 2 , ,B k }. Then, by definition, there is a set {i 1 ,i 2 , ,i k } with 1 ≤ i 1 <i 2 < ···<i k ≤ m such that B 1 is the component of T(S i k ,α i k ) ∩ T(S i 1 ,α i 1 ) containing x, and such that B j , j = 2,3, ,k is the component of T(S i j−1 ,α i j−1 ) ∩ T(S i j ,α i j ) containing s i j−1 ◦ s i 2 ◦ s i 1 (x). If there is a B j is of the form B (∗,i j ;∗) , that is, a component which is not near to α i j ,then B j = B (i k ,i 1 ;∗) for j = 1; B j = B (i j−1 ,i j ;∗) for j = 1. When j<k,wehavethats i j ◦ s i j−1 ◦···◦s i 1 (x) ∈ s i j (B j ) ∈ B j+1 .Sinces i j (B (∗,i j ;∗) )does not meet any component of the form B (∗,i j ;∗) but itself, B j+1 = B (i j ,∗;∗) . Because B j+1 lies in T(S i j+1 ,α i j+1 ), we have B j+1 = B (i j ,i j+1 ;∗) . Similarly, we can prove that B 1 = B (i k ,i i ;∗) if B k = B (i k−1 ,i k ;∗) . Notice that each B j is only one of two types: either B j = B (i j−1 ,i j ;∗) or B (i j ,i j+1 ;∗) .The above arguments have shown that if one component B j in an accompanying sequence is of the first type, the others are the same as it. Thus, we are done.  [...]... implies that the components in one accompanying sequence are distinct The next lemma will show that after some suitable isotopies on the slide homeomorphism, there is a one-to-one correspondence between the fixed point set on the sliding set and the set consisting of the above accompanying sequences Lemma 4.4 If the sliding set ∪m 1 T(S j ,α j ) is in general position, we can isotope the slide j= homeomorphisms. .. point of f is of one of two types listed there Step 2 Fixed points having accompanying sequences of the first type Consider a sequence {B(ik ,i1 ; j1 ) ,B(i1 ,i2 ; j2 ) ,B(i2 ,i3 ; j3 ) , ,B(ik−1 ,ik ; jk ) } of the components of the intersection of sliding sets Since B(ik ,i1 ; j1 ) ranges in t-direction from one component of ∂T(Si1 ,αi1 ) to the other component of ∂T(Si1 ,αi1 ), its image under si1 will... immediately our conclusion By this proposition, the number |αi ∩ S j | determines in some sense the number of fixed points In order to reduce the number of fixed points of such homeomorphisms, the intersection numbers (|αi ∩ S j |’s) should be reduced In [9, page 184], we defined α ∩ Sj : α MI αi ,S j =: min α j rel{0,1}, α has no self intersection (4.11) From this definition, we have Proposition 4.7 Let S j be... ) will satisfy the condition: one eigenvalue has absolute value greater than 1, the other two have absolute values less than 1 We assume that |λ1 | > 1, |λ2 | < 1 and |λ3 | < 1 From Figure 4.3, we know that the θ-direction of B(i, j;k) is mapped by s j into the θdirection of B( j,∗;∗) if I(i, j;k) > 0; the θ-direction of B(i, j;k ) is mapped by s j into opposition of the θ-direction of B( j,∗;∗) if... longer a composition of standard slide homeomorphisms s j , we still say that {B1 ,B2 , ,Bk } is the “accompanying sequence” of x in the following sense: B1 is the component of T(Sik ,αik ) ∩ T(Si1 ,αi1 ) containing x, B j , j = 2,3, ,k is the component of T(Si j −1 ,αi j −1 ) ∩ T(Si j ,αi j ) containing si j −1 ◦ · · · ◦ si2 ◦ si1 (x) The proof of Lemma 4.1 still works as long as s j and s j have the. .. works as long as s j and s j have the same support set T(S j ,α j ) for each j Proof of Lemma 4.4 We will give the proof in three steps Step 1 Isotope each s(S j ,α j ) 12 Fixed points of slide homeomorphisms Consider an arbitrary component B(i, j;k) of the intersection of the sliding sets Since it is a component near the kth point in αi ∩ S j , we can assume that B(i, j;k) ⊂ ci−1 (θ,ϕ,t) : θ − θ(i,... has unique fixed point on above set Thus, the fixed point of f with accompanying sequence {B1 ,B2 , ,Bk } is unique Let x∗ be the unique fixed point of f in the set in (4.9) Then ci1 (x∗ ) is a fixed point of ci1 ◦ f ◦ ci−1 : U → ci1 (T(Si1 ,αi1 )), where U is the ci1 image of the set in (4.9) Using 1 the coordinates of T 2 × I, the three eigenvalues λ1 , λ2 , λ3 of the derivative of ci1 ◦ f ◦ ci−1 1 at... that there is a unique fixed point y∗ of f having {B(i1 ,il ;∗) ,B(i2 ,i1 ;∗) ,B(i3 ,i2 ;∗) , ,B(il ,il−1 ;∗) } as its accompanying sequence The only difference is in the fixed point index Because the three eigenvalues λ1 , λ2 , λ3 of the derivative of ci1 ◦ ( f )−1 ◦ ci−1 at ci1 (y∗ ) satisfy the conditions: |λ1 | > 1, |λ2 | < 1 and 1 |λ3 | < 1, the three eigenvalues μ1 , μ2 , μ3 of the derivative of ci1... ,il−1 ; jl ) This lemma is a generalization of [9, Lemma 4.2] The proof here is more descriptive than the direct computation there The fixed point class coordinates of these fixed points can be computed in the same way Proposition 4.6 Let f = s(Sm ,αm ) ◦ s(Sm−1 ,αm−1 ) ◦ · · · ◦ s(S1 ,α1 ) be a homeomorphism composed of finitely many slide homeomorphisms Assume that the S j ’s are pairwise disjoint, Xuezhi... also the fixed point index ind( f ,x∗ ) by the commutativity of fixed point index Step 3 Fixed points having accompanying sequences of the second type 14 Fixed points of slide homeomorphisms αi αj Sj q(i, j;k B( j,∗;∗) ) q(i, j;k) αj s j (B(i, j;k) ) B(i, j;k) αi Figure 4.3 ¯ ¯ ¯ Note that the inverse ( f )−1 = s1 ◦ s2 ◦ · · · ◦ sm of f is also a homeomorphism com¯ posed of finite isotoped slide homeomorphisms, . the Nielsen numbers and fixed points of homeomor- phisms which are compositions of m slide homeomorphisms on nonprime 3-manifolds. The fixed point index of the complement of the union of the sliding. immediately our conclusion.  By this proposition, the number |α i ∩ S j | determines in some sense the number of fixed points. In order to reduce the number of fixed points of such homeomorphisms, the intersection. be the accom- panying sequence of x in the components of the intersection of sliding sets. Clearly, the set {B 1 ,B 2 , ,B k } itself is just the set of all components of the intersection of